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Realization of nonlinear behaviors from organizing centers

Alessio Franci, Rodolphe Sepulchre

arXiv:1409.5560v1math.OC

TL;DR

The paper addresses how prescribed nonlinear behaviors can be realized as internal circuit structures. It uses singularity organizing centers and monotone feedback interconnections to construct static and dynamic behaviors, recovering switches, relaxation oscillators, rest-spike models, and bursters within a restricted framework.

  • Problem

    The paper addresses the system-theory problem of realizing prescribed external behaviors as minimal internal systems.

    Source: §I · p. 1

  • Method

    The method identifies singularity organizing centers, realizes their unfoldings with monotone feedback interconnections, and adds dynamics whose limit sets reflect the static behavior.

    Source: §I · p. 1, §III · p. 2

  • Results

    The restricted framework recovers bistable switches and relaxation oscillators from hysteresis, and rest-spike bistable models and bursters from the winged-cusp.

    Source: §I · p. 1

  • Takeaways & Limitations

    The circuit interpretation emphasizes feedback and monotonicity in organizing nonlinear behaviors and may aid analysis and synthesis of tunable, robust nonlinear devices.

    Source: §V · p. 6

    The exposition covers only hysteresis and winged-cusp singularities, first-order linear filters, and singular-perturbation-based inference of dynamics from static analysis.

    Source: §I · p. 1

Abstract

from arXiv · show

Borrowing the concept of organizing center from singularity theory, the paper proposes a methodology to realize nonlinear behaviors such as switches, relaxation oscillators, or bursters from core circuits that reveal the fundamental role of monotonicity and feedback in their robustness and modulation.

I. INTRODUCTION

The paper develops a way to realize robust nonlinear behaviors from singularities, using monotone feedback circuits and extending static behavior into dynamics. It illustrates the approach with hysteresis and winged-cusp organizing centers, while restricting the exposition to two singularities and first-order filters.

I. INTRODUCTION

  • Robust static behaviors are represented as local deformations of a singular behavior in unfolding parameters, whose number is the singularity's codimension.The organizing center is identified by local algebraic conditions in a normal form.Source: §I · p. 1
  • The algebraic characterization of a singularity and its unfolding translates into feedback interconnections of monotone nonlinearities.The interconnection laws are parameterized by unfolding parameters, exposing how monotonicity and feedback organize static behavior.Source: §I · p. 1
  • The construction turns singularity-organized static behaviors into nonlinear dynamics whose limit sets are shaped by the static behavior.The hierarchy among state, bifurcation parameter, and unfolding parameters becomes a hierarchy of time-scales; singular perturbation theory establishes nearby hyperbolic attractors.Source: §I · p. 1
  • The exposition restricts the general procedure to hysteresis and winged-cusp singularities, first-order linear filters, and singular-perturbation-based analysis.This setup yields multiple-time-scale nonlinear behaviors.Source: §I · p. 1
  • The method recovers bistable switches and relaxation oscillators from hysteresis, and rest-spike bistable models and bursters from the winged-cusp.Its motivation includes prior work associating winged-cusp organization in biophysical neuron models with physiological regulatory feedback parameters that modulate bursters [5].Source: §I · p. 1

II. SINGULARITIES ORGANIZE STATIC BEHAVIORS

  • Singularity theory organizes robust static behaviors into families associated with a singularity and its universal unfolding.Unfolding parameters shape the input-controlled behavior, and transition varieties separate qualitatively distinct regions of the unfolding space.Source: §II · p. 2, Figure 1 · p. 2

III. FROM UNIVERSAL UNFOLDINGS TO STATIC CIRCUITS

Singular behaviors and their universal unfoldings can be represented as interconnections of monotone sigmoidal nonlinearities. This circuit representation makes the organizing structure of static behaviors identifiable from interconnection structure.

III. FROM UNIVERSAL UNFOLDINGS TO STATIC CIRCUITS

  • Singular behaviors and their universal unfoldings can be realized as interconnections of monotone sigmoidal nonlinearities.The circuit characterization helps identify organizing centers of static behaviors from interconnection structures.Source: §III · p. 2
  • The circuit interpretation links algebraic descriptions of singular behavior to interconnections of these nonlinearities.The paper introduces derivative notation for smooth functions in the course of this construction.Source: §III · p. 2
  • In this paper, a sigmoidal nonlinearity is odd, monotone, saturated, and has specified nonzero odd-order derivatives at zero.The stated conditions include S'(0) = 1; examples include sigmoids, saturations, and hyperbolic tangents.Source: §III · p. 3

A. A static circuit realization of the hysteresis

A positive-feedback loop around a sigmoid realizes the hysteresis singularity, and adding β gives its universal unfolding. The unfolding parameter organizes the persistent static behaviors.

A. A static circuit realization of the hysteresis

  • At β = 0, the static feedback behavior −y + S(y + u) = 0 is strongly equivalent to the hysteresis singularity.The paper presents the recognition proof as an elementary application of sigmoidal properties.Source: A. A static circuit realization of the… · p. 3
  • Adding βy inside the sigmoid produces a universal unfolding of the hysteresis, with β as its unfolding parameter.Its persistent static behaviors are given by the persistent bifurcation diagrams of the hysteresis.Source: A. A static circuit realization of the… · p. 3

B. The winged-cusp static behavior by interconnection

The winged-cusp behavior is constructed by combining the hysteresis circuit with a non-monotone bump derived from monotone nonlinearities. The resulting circuit realizes the winged cusp and its universal unfolding.

B. The winged-cusp static behavior by interconnection

IV. FROM STATIC TO DYNAMIC BEHAVIORS

The paper adds first-order filters to static feedback interconnections to obtain dynamic behaviors. The singularity hierarchy maps to separated time scales, and singular perturbation theory supports persistence of the static organizing structure.

IV. FROM STATIC TO DYNAMIC BEHAVIORS

  • Adding first-order filters to static feedback interconnections provides the paper’s construction of dynamic behaviors.The qualitative results are stated to persist when the filters are replaced by arbitrary monotone systems.Source: §IV · p. 4
  • The hierarchy among state, bifurcation, and unfolding variables dictates corresponding fast, slow, and ultra-slow time scales.Singular perturbation theory ensures persistence of the singular structure when the time scales are sufficiently separated; monotone filters preserve the organizing singularity.Source: §IV · p. 4
  • Detailed proofs for this construction are referred to the authors’ recent paper [5].Source: §IV · p. 4

1) Bistability:

A first-order filter added to the hysteresis feedback loop gives a dynamical model of bistability. Its possible behavior is predicted from the static behavior.

1) Bistability:

  • The model is described as reminiscent of autocatalysis and switching behaviors, including the Hopfield neuronal, fast Hodgkin-Huxley, and Griffith genetic-switch models.Source: 1) Bistability · p. 4
  • For the figure’s β > 0 sketch, solid branches denote stable steady states, dashed branches unstable states, and F marks a fold; the pulse response uses β = 0.5.Source: Figure 5 · p. 4
  • For β > 0, the filtered model exhibits persistent bistability, and the same conclusions apply with an arbitrary monotone system replacing the linear filter.The model uses a first-order filter with transfer function Hf(s) := 1/(εf s + 1), where 0 < εf ≪ 1.Source: 1) Bistability · p. 4

2) Relaxation oscillations and excitability:

A slow negative-feedback loop modulates a fast hysteretic circuit, producing relaxation oscillations and excitable responses. The critical manifold links the circuit’s static hysteresis to its slow-fast dynamics.

2) Relaxation oscillations and excitability:

  • A slow negative-feedback loop modulates the fast autocatalytic loop, shifting the hysteresis singularity to β = 1.The circuit adds slow feedback to a bistable switch.Source: 2) Relaxation oscillations and… · p. 4
  • The critical manifold is the static hysteresis with control parameter ũ = u − xs, so its unfolding captures the dynamics in the singular limit.For β > 0, the multivalued manifold and slow negative feedback transform fast bistability into relaxation oscillations.Source: 2) Relaxation oscillations and… · p. 4
  • For 0 < β < 1 and sufficiently small εf and |u|, the system has one exponentially unstable steady state surrounded by an almost globally exponentially stable relaxation limit cycle.The cycle is organized by the hysteresis critical manifold.Source: 2) Relaxation oscillations and… · pp. 4–5, Figure 6 · p. 5
  • The relaxation-cycle result also holds when first-order filters are replaced by monotone systems.The phase-plane construction describes the cycle around the bistable static behavior.Source: 2) Relaxation oscillations and… · p. 5
  • In the excitable regime, small pulses can evoke large transient responses; the spike reflects fast-subsystem bistability and is transient because of slow adaptation.As u varies, a Hopf bifurcation can leave a unique exponentially stable steady state.Source: 2) Relaxation oscillations and… · p. 5, Figure 6 · p. 5

1) Rest-spike bistability:

A winged-cusp-organized mirrored hysteresis supports rest-spike bistability. Under the stated parameter conditions, perturbing α across the transcritical bifurcation can switch rest to periodic spiking.

1) Rest-spike bistability:

  • The analysis fixes β and γ while varying α around the transcritical transition variety in the winged cusp’s universal unfolding.Figure 7 depicts the circuit, phase portrait, and response to a transient α perturbation.Source: 1) Rest-spike bistability · p. 5, Figure 7 · p. 5
  • The mirrored hysteresis persistent bifurcation diagram organizes the rest-spike bistable behavior for suitable unfolding parameters.The suitable parameters depend on the specific functional form of S(·).Source: 1) Rest-spike bistability · p. 5
  • For β > 0, near the pitchfork singularity and with sufficiently small εf, an exponentially stable fixed point coexists with an exponentially stable relaxation limit cycle.Their attraction basins are separated by the stable manifold of a hyperbolic saddle.Source: 1) Rest-spike bistability · p. 5
  • Under Theorem 2’s conditions, a short perturbation across the transcritical bifurcation variety can switch the system from rest to periodic oscillations.The perturbation is applied transiently to unfolding parameter α.Source: 1) Rest-spike bistability · p. 5

2) Bursting:

An ultra-slow negative-feedback loop uses winged-cusp modulation to turn rest-spike bistability into bursting. The paper also describes a control-parameter transition from bursting to tonic spiking and frames feedback and monotonicity as organizing principles for nonlinear behaviors.

2) Bursting:

  • An ultra-slow negative-feedback loop transforms rest-spike coexistence into a three-timescale monostable bursting behavior.The unfolding parameter α modulates the winged cusp monotonically and serves as an adaptation parameter.Source: 2) Bursting · p. 5, Figure 8 · p. 6
  • For β > 0, suitable control and unfolding parameters, and sufficiently small εf and εu, the circuit possesses a hyperbolic bursting attractor.The theorem specifies parameters near the pitchfork singularity in the winged cusp’s universal unfolding.Source: 2) Bursting · p. 6
  • Increasing the control parameter beyond the rest-spike coexistence region shifts behavior from bursting toward slow spiking at a transcritical-homoclinic bifurcation.The transition occurs when the saddle-node and homoclinic bifurcations delimiting rest-spike bistability merge.Source: 2) Bursting · p. 6, Figure 8 · p. 6

V. CONCLUSIONS

  • The proposed circuit interpretation highlights feedback and monotonicity in organizing nonlinear behaviors and may aid analysis and synthesis of tunable, robust nonlinear devices.The paper presents the methodology as a way to identify core circuit organization in biological signalling.Source: §V · p. 6
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